Projectively and Conformally Invariant Star-Products

We consider the Poisson algebra S(M) of smooth functions on T ^* M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1,) (resp. SO(p+1,q+1))-invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.     

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A_∞-algebras) by closed strings (L_∞-algebras). H. K is supported by JSPS Research Fellowships for Young Scientists. J. S. is supported in part by NSF grant FRG DMS-0139799 and US-Czech Republic grant INT-0203119.     

Polynomial Realization of <Emphasis Type="Italic">s</Emphasis>l<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2) and Fusion Rules at Exceptional Values of <Emphasis Type="Italic">q</Emphasis>

Representations of the sℓ_q(2) algebra are constructed in the space of polynomials of real (complex) variable for q^N=1. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.     

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK ²ⁿ is raised.In memory of Julian Schwinger     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

A convergent star product for linear Poisson structures

An explicit star product ⋆ _α ^Γ on the dual of a general Lie algebra equipped with the linear Poisson bracket is constructed. An equivalence operator between this star product and the Kontsevich star product in [K₁] is given and diverse properties of the star product ⋆ _α ^Γ are studied. It is also proved that the star product ⋆ _α ^Γ provides a convergent deformation quantization in the sense of Rieffel [R₁].     

Deformation Quantizations of the Poisson Algebra of Laurent Polynomials

It is well known that the Moyal bracket gives a unique deformation quantization of the canonical phase space R2n up to equivalence. In his presentation of an interesting deformation quantization of the Poisson algebra of Laurent polynomials, Ovsienko discusses the equivalences of deformation quantizations of these algebras. We show that under suitable conditions, deformation quantizations of this algebra are equivalent. Though Ovsienko showed that there exists a deformation quantization of the Poisson algebra of Laurent polynomials which is not equivalent to the Moyal product, this is not correct. We show this equivalence by two methods: a direct construction of the intertwiner via the star exponential and a more standard approach using Hochschild 2-cocycles.     

Method of Quantum Characters in Equivariant Quantization

 Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose 𝔸 _h (M) is a 𝕌 _h (g)-equivariant quantization of the function algebra 𝔸(M) on M. We develop a method of building 𝕌 _h (g)-equivariant quantization on G-orbits in M as quotients of 𝔸 _h (M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in 𝕌^* _h (g) and quotients of 𝔸 _h (M). It turns out that they are in one-to-one correspondence with characters of the algebra 𝔸 _h (M). We specialize our approach to the situation g=gl(n,ℂ), M=End(ℂ ⁿ ), and 𝔸 _h (M) the so-called reflection equation algebra associated with the representation of 𝕌 _h (g) on ℂ ⁿ . For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the 𝕌(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits. Received: 28 April 2002 / Accepted: 3 October 2002 Published online: 24 January 2003 RID="*" ID="*" This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01. Communicated by L. Takhtajan     

Noncommutative Deformation Theory

In Gerstenhaber's classical theory of deformations, the deformation parameter commutes with the original algebra. Motivated by some non classical deformations which recently appeared for quantization of Nambu mechanics, we introduce new deformations where the parameter no longer commutes with the original algebra. We find the associated cohomology and Gerstenhaber algebra and give rigidity and integrability criterions. We show that the Weyl algebra (though rigid in classical theory) can be nontrivially deformed, in super-commutative theory, to the supersymmetry enveloping algebra      

Remarks on the quantization of conformal fields

The quantization of a general (b, c) system in two dimensions is formulated in terms of an infinite hierarchy of modules for the Virasoro algebra that interpolate between the space of classical conformal fields of weight j and the Dirac sea of semi-infinite forms. This provides a natural framework in which to study the relation between algebraic geometry and representations of the Virasoro algebra with central charge c_j=−2(6j²−6j+1). The importance of the construction is discussed in the context of string theory.     

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

 S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C ^* -algebra of . The proofs of all these results depend on various properties of q-hypergeometric ₁ϕ₁ functions. Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 RID="*" ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.) Communicated by L. Takhtajan     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. II

We discuss some properties of a non-commutative generalization of the classical moment problem (them-problem) previously introduced. It is shown that there is a connexion between the determination of the problem and the self-adjointness properties in the corresponding Hilbert space. This generalizes the well-known connexion between the determination of the measure in the classical moment problem and the self-adjointness properties of the polynomials as operators in the correspondingL ²-space. The dependence of them-problem on the choice ofC*-semi-norms and on the action of *-homomorphisms is also investigated. As an application, it is shown that if a quantum field (in a very general sense) is essentially self-adjoint then them-problem for the Wightman functional is determined on the quasi-localizableC*-algebra and that the corresponding representation of the localizable algebra generates the bounded observables of the field. It is pointed out that (ultraviolet and spatially) cut-off fields fall in this class and, therefore, are in one to one correspondance with states on the quasi-localizableC*-algebra.Laboratoire associé au Centre National de la Recherche Scientifique.     

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL ₂ . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL ₂ , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra. Received: 20 April 1997 / Accepted: 22 July 1997     

Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.     

Non-𝒜belian Geometry

 Spatial noncommutativity is similar and can even be related to the non- Abelian nature of multiple D-branes. But they have so far seemed independent of each other. Reflecting this decoupling, the algebra of matrix valued fields on noncommutative space is thought to be the simple tensor product of constant matrix algebra and the Moyal-Weyl deformation. We propose scenarios in which the two become intertwined and inseparable. Therefore the usual separation of ordinary or noncommutative space from the internal discrete space responsible for non-Abelian symmetry is really the exceptional case of an unified structure. We call it non-Abelian geometry. This general structure emerges when multiple D-branes are configured suitably in a flat but varying B field background, or in the presence of non-Abelian gauge field background. It can also occur in connection with Taub-NUT geometry. We compute the deformed product of matrix valued functions using the lattice string quantum mechanical model developed earlier. The result is a new type of associative algebra defining non-Abelian geometry. A possible supergravity dual is also discussed. Received: 13 December 2000 / Accepted: 24 October 2002 Published online: 24 January 2003 Communicated by R. H. Dijkgraaf     

Deformation Quantization on the Closure of Minimal Coadjoint Orbits

We consider a complex simple Lie algebra ${\mathfrak{g}}$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct ${\mathfrak{g}}$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra ${{\rm U}(\mathfrak{g})}$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.